Parts (REF ), (REF ) and (REF ) for \(\overline{\mathcal {J}}_{g,n}(\phi )\) , in the special case \(n=1\)  and for specific choices of \(\phi \in V_{g,n}\) , have been proved by Abreu-Pacini in [1]}. Both their proof and ours follow the blueprint provided by [2]} and go by giving an explicit description of the toroidal stratification (see Proposition REF  above).
